# transformations replaced by symmetry

♦
8 matching pages ♦

(0.002 seconds)

## 8 matching pages

##### 1: 19.15 Advantages of Symmetry

…
►Symmetry in $x,y,z$ of ${R}_{F}(x,y,z)$, ${R}_{G}(x,y,z)$, and ${R}_{J}(x,y,z,p)$
replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17).
…
…

##### 2: 19.22 Quadratic Transformations

…
►
…

##### 3: 19.25 Relations to Other Functions

…
►

19.25.16
$$\mathrm{\Pi}(\varphi ,{\alpha}^{2},k)=-\frac{1}{3}{\omega}^{2}{R}_{J}(c-1,c-{k}^{2},c,c-{\omega}^{2})+\sqrt{\frac{(c-1)(c-{k}^{2})}{({\alpha}^{2}-1)(1-{\omega}^{2})}}{R}_{C}(c({\alpha}^{2}-1)(1-{\omega}^{2}),({\alpha}^{2}-c)(c-{\omega}^{2})),$$
${\omega}^{2}={k}^{2}/{\alpha}^{2}$.

…
##### 4: Bille C. Carlson

…
►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few.
…

##### 5: Errata

…
►
Section 24.1
…
►
Chapter 18 Orthogonal Polynomials
…
►
Figure 4.3.1
…
►
Section 27.20
…
►
Subsection 21.10(i)
…

The text “greatest common divisor of $m,n$” was replaced with “greatest common divisor of $k,m$”.

This figure was rescaled, with symmetry lines added, to make evident the symmetry due to the inverse relationship between the two functions.

*Reported 2015-11-12 by James W. Pitman.*

The entire Section was replaced.

The entire original content of this subsection has been replaced by a reference.

##### 6: 18.7 Interrelations and Limit Relations

…
►

###### §18.7(i) Linear Transformations

… ►###### §18.7(ii) Quadratic Transformations

… ►
18.7.17
$${U}_{2n}\left(x\right)={W}_{n}\left(2{x}^{2}-1\right),$$

►
18.7.18
$${T}_{2n+1}\left(x\right)=x{V}_{n}\left(2{x}^{2}-1\right).$$

…
##### 7: 18.17 Integrals

…
►and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1.
…
►

###### §18.17(v) Fourier Transforms

… ►###### Jacobi

… ►###### §18.17(vi) Laplace Transforms

… ►###### §18.17(vii) Mellin Transforms

…##### 8: 20.11 Generalizations and Analogs

…
►If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)):
…
►